|
%I #13 Apr 16 2022 17:29:37
%S 1,8,28,64,126,416,1680,5248,13973,53008,224092,768128,2298362,
%T 9091712,38646080,138919680,445239890,1791759736,7578341516,
%U 28075757440,94326701336,382678126688,1606044597904,6077887574016,21141924364377,86127882138832,358591678307792
%N G.f. A(x) satisfies: A(x) = ( A(-x) + sqrt( A(-x)^2 + 64*x/A(-x) ) )/2.
%C Equals the self-convolution 8th power of A353324.
%H Paul D. Hanna, <a href="/A353325/b353325.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
%F (1) A(x) = ( A(-x) + sqrt( A(-x)^2 + 64*x/A(-x) ) )/2.
%F (2) A(-x) = ( A(x) + sqrt( A(x)^2 - 64*x/A(x) ) )/2.
%F (3) A(x)*A(-x) * (A(x) - A(-x)) = 16*x.
%F (4) A(-x) = (2 - A(x)^(1/8))^8.
%F (5) A(x) = (1 + (A(x)^(1/8) - A(-x)^(1/8))/2)^8.
%F (6) A(x) = ( 1 + Series_Reversion( x*(1 - x^2)^8*(1 + x^2)*(1 + 6*x^2 + x^4) ) )^8.
%e G.f.: A(x) = 1 + 8*x + 28*x^2 + 64*x^3 + 126*x^4 + 416*x^5 + 1680*x^6 + 5248*x^7 + 13973*x^8 + 53008*x^9 + 224092*x^10 + ...
%e such that A(-x) = ( A(x) + sqrt( A(x)^2 - 64*x/A(x) ) )/2.
%e Related series.
%e A(x)^(1/8) = 1 + x + x^3 + 24*x^5 + 95*x^7 + 2699*x^9 + 15803*x^11 + 426524*x^13 + 3152930*x^15 + 78893000*x^17 + ... + A353324(n)*x^n + ...
%e sqrt( A(x)^2 - 64*x/A(x) ) = 1 - 24*x + 28*x^2 - 192*x^3 + 126*x^4 - 1248*x^5 + 1680*x^6 - 15744*x^7 + 13973*x^8 - 159024*x^9 + ...
%e A(x)*A(-x) = 1 - 8*x^2 + 12*x^4 - 336*x^6 + 686*x^8 - 38896*x^10 + 48064*x^12 - 6011040*x^14 + 132853*x^16 - 1072398368*x^18 + ...
%o (PARI) {a(n) = my(A=1+8*x+28*x^2,B=1-8*x+28*x^2); for(i=0,n,
%o A = (B + sqrt(B^2 + 64*x/B +x*O(x^n)))/2;
%o B = (A + sqrt(A^2 - 64*x/A +x*O(x^n)))/2;
%o B = (1 - (A^(1/8) - B^(1/8))/2)^8; );
%o polcoeff(A,n)}
%o for(n=0,20,print1(a(n),", "))
%o (PARI) /* Series Reversion Formula */
%o {a(n) = my(A = (1 + serreverse(x*(1 - x^2)^8*(1 + x^2)*(1 + 6*x^2 + x^4) +x^2*O(x^n) ))^8); polcoeff(A,n)}
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A353324, A353326.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Apr 13 2022
| 